Let $A, B$ be two structures and $f : A \to B$ an embedding, that is, an injective strong homomorphism. Does this imply that $A$ is isomorphic to a substructure of $B$?
Also: Does being isomorphic to a substructure of $B$ for $A$ imply that there is an embedding $A\to B$?
Yes, those two properties are equivalent. If $f$ is an embedding of $A$ into $B$, then $A$ is isomorphic to its image under $f$, which is a substructure of $B$; conversely, if $A$ is isomorphic to $A'\subseteq B$, then any isomorphism $i$ is also an embedding of $A$ into $B$. Both parts are easily checked.